To find the mode from a grouped frequency distribution, we use the formula for the mode in a continuous frequency distribution:

$\text{Mode} = L + \left( \frac{f_m - f_{m-1}}{2f_m - f_{m-1} - f_{m+1}} \right) \times h$

where:

• $L$ is the lower limit of the modal class.
• $f_m$ is the frequency of the modal class.
• $f_{m-1}$ is the frequency of the class before the modal class.
• $f_{m+1}$ is the frequency of the class after the modal class.
• $h$ is the class interval width.

Analyzing the Table

From your uploaded image, the class intervals and frequencies are as follows:

Identifying the Modal Class

The modal class is the class interval with the highest frequency. Here, the highest frequency is 27, corresponding to the class interval 31-38.

Applying the Formula

• $L = 31$ (the lower limit of the modal class).
• $f_m = 27$ (frequency of the modal class).
• $f_{m-1} = 23$ (frequency of the class before the modal class).
• $f_{m+1} = 13$ (frequency of the class after the modal class).
• $h = 7$ (class width, assuming each interval has the same width).

Substitute these values into the formula:

$\text{Mode} = 31 + \left( \frac{27 - 23}{2 \times 27 - 23 - 13} \right) \times 7$

Calculating further:

1. The numerator is $27 - 23 = 4$.
2. The denominator is $2 \times 27 - 23 - 13 = 54 - 23 - 13 = 18$.

$\text{Mode} = 31 + \left( \frac{4}{18} \right) \times 7$

3. Simplify $\frac{4}{18} = \frac{2}{9}$.

$\text{Mode} = 31 + \left( \frac{2}{9} \right) \times 7$

4. $\frac{2}{9} \times 7 = \frac{14}{9} \approx 1.56$.

$\text{Mode} \approx 31 + 1.56 = 32.56$

Answer

The mode is approximately 32.56.

Would you like further details or have any questions?

Here are some related questions for further exploration:

1. How does the mode compare to the mean in grouped data?
2. What if two classes had the same highest frequency? How would that affect the mode?
3. How is the mode used in real-world data analysis?
4. Can the mode be calculated differently for ungrouped data?
5. How does the mode relate to the median in skewed distributions?

Tip: When analyzing grouped data, ensure all intervals are of equal width, as this consistency is essential for accurate mode calculations.